Is the work on Fermat's Theorem really done?

[dkotschessaa]

I'm still relatively new to mathematics, in the sense of studying it with any degree of seriousness, so I have a question related to the general field of mathematics and a little bit on it's history.

I haven't read Simon Singh's book yet,but a I understand the story on Fermat's Last Theorem, Fermat himself allegedly had a proof which was lost, puzzling mathematicians for many years who attempted to find the proof. The work seems to have been abandoned by most people for a number of years, only to be solved (generally for n) finally by Andrew Wiles in 1994 (after correcting a failed 1993 attempt) who worked in secrecy for about 7 years.

But from what I gather, the proof was extremely long and complicated, using a number of mathematical tools developed by others, and as such is a thoroughly "modern" proof - not what Fermat himself would have used. So is any work on this still being done to find something closer to Fermat's original proof, or is it believed there was no such proof, or is it simply left at Wiles? Can anybody say anything on the "elegance" of Wiles proof? It sounds like a monster to me but I haven't seen it nor would I have any chance of doing so just yet.

-DaveKA

 

[JCVD]

It is generally accepted that Fermat did not have a correct proof himself. He famously wrote his conjecture in the margin of a copy of a text he was reading, but wrote that even though he had a proof the margin was too small for him to write it there. He did have a proof for exponents equal to 4, which reduced the problem to considering only prime exponents. He may have thought some sort of inductive argument would work but didn't feel like working out the details.

 

[dkotschessaa]

It is generally accepted that Fermat did not have a correct proof himself. He famously wrote his conjecture in the margin of a copy of a text he was reading, but wrote that even though he had a proof the margin was too small for him to write it there. He did have a proof for exponents equal to 4, which reduced the problem to considering only prime exponents. He may have thought some sort of inductive argument would work but didn't feel like working out the details.

I wonder if he had any idea what he'd started or how complicated it would be... It seems he thought it would be a lot simpler. That equation looks so innocent.

 

[said_alyami]

about Wiles proof, there is a real problem about it: its too complicated to be verified by most mathematicians. in other words, its believed that only few people in the world can investigate it.

I am sure Wiles is a great mathematician with good intentions,, but when something is too complex to be understood, the devil finds great corners and even mansions to hide in.

its something close to what Wall street guys did when they have invented the complicated financial "derivatives", it was based on mathematical models they believed were so tight they were given AAA rating.

there models were so complicated and "not understood well" to the degree other people including professionals thought its only those guys are so smart, and they were just too stupid to understand them.
I saw an interview with Alan Greinspan, man on the Fed reserve for 14 years, he said: i could not understand those derivatives and I have a degree in math!

Finally after the financial meltdown, it was then so clear those derivatives did not even deserve the ink and papers they were wrote on. they were a disaster, and the tight mathematical models they thought were so tight was proven to be big fallacies.

Moral of the story ,, when something is too complicated to be verified ,,, you should be really skeptical. and this is what i feel about Wiles proof.

on 1993 he believed he had the perfect proof till a colleague found a problem in his proof, then he worked on it again to bring the final proof.
May be he just made it more complicated that even that colleague and Wiles himself could not see anything anymore, although it might be still there.

this is my opinion anyway.

 

[AlephZero]

Read Singh's book.

A short answer to your question is that the Wiles proof of FLT is a (rather trivial) deduction from a whole new area of math that has been opened up, and won't be "finished" for a very long time (I would predict for decades, if not centuries).

In that sense, FLT it turned out to be a very different type of problem from the 4-color map problem, where the computer-assisted proof with thousands of special cases didn't really add much to the "big picture" of math at all, though it may have helped to advance the technology for creating computer-assisted proofs.

The Wiles proof doesn't tell us anything about Fermat's proof (if he had one), since there is no feasible way Fermat could have invented the Wiles' proof given the development of math when he was living. The history of math has plenty of occasions where an initial complex proof is later replaced by a much simpler one. But if I was betting on this, I would predict the "simpler proof" would again be a result of more breakthroughs in the general theory behind Wiles' proof, not a "one-off" argument that just proves FLT.

 

[bapowell]

@said_alyami: So what you're saying is that unless a solution is simple it's effectively wrong?

 

[DaveC426913]

@said_alyami: So what you're saying is that unless a solution is simple it's effectively wrong?

I think he's simply saying: until a solution can be independently verified, it cannot be said to be a valid solution.

Substitute the word ''theory' for 'solution' and you have garden-variety science methodology.

 

[said_alyami]

I think he's simply saying: until a solution can be independently verified, it cannot be said to be a valid solution.

Substitute the word ''theory' for 'solution' and you have garden-variety science methodology.

Exactly!

 

[dkotschessaa]

Maybe I should stop asking and get to my book, but I like hearing everybody's opinion here.

Are people still working with FLT? It seems like just the work itself would be rewarding, even if you aren't exactly looking for a solution. Someone with basic algebra could probably at least try and see where it takes them.

-DaveKA

 

[discrete*]

I'd like to believe that Fermat had a very nice and elegant proof stashed away -- some clever thing that no one has thought of -- but logic just won't allow it. In reality, it seems nearly impossible that Fermat had a proof that would be acceptable by today's standard of rigor.

I've heard a Professor say exactly what said_alyami did; that the proof is too complicated to be understood by the mathematical community. I'm not sure if this is a testament to Wiles's genius, or to a sort of madness.

To answer your original question: No, I do not believe work on Fermat's theorem is really done. I sincerely believe that some day a mathematician somewhere will create a much "simpler" proof of the theorem, or disprove it outright. The same for Goldbach's Conjecture.

 

[Char. Limit]

Maybe I should stop asking and get to my book, but I like hearing everybody's opinion here.

Are people still working with FLT? It seems like just the work itself would be rewarding, even if you aren't exactly looking for a solution. Someone with basic algebra could probably at least try and see where it takes them.

-DaveKA

My friend's grandfather is a mathematician who's trying to find a simple proof to FLT. So it's still going on.

 

[robert Ihnot]

The idea, long cherished by amateurs that a really simple proof is laying around somewhere doesn't stack up. In 1908 industrialist Paul Wolfskehl offered a 10,000 mark prize for the solution. This generated 621 incorrect proofs in the first year, though the number slowed down after that. With the change in Germany's financial condition the prize money went broke, but was restored in later years, and Wiles was awarded $50,000 for his solution.

Most of the solution submitted over the years for the prize money were of a very simple type that was the mark of an amateur.

Howard Eves said, "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published."

 

[statdad]

"I saw an interview with Alan Greinspan, man on the Fed reserve for 14 years, he said: i could not understand those derivatives and I have a degree in math!"

Alan Greenspan does not have a mathematics degree: he said that neither he nor some of the best mathematicians could understand the financial instruments that had been developed.